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L‐curve analysis of radiotherapy optimization problems
Author(s) -
Chvetsov Alexei V.
Publication year - 2005
Publication title -
medical physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.473
H-Index - 180
eISSN - 2473-4209
pISSN - 0094-2405
DOI - 10.1118/1.1949750
Subject(s) - regularization (linguistics) , mathematics , curvature , inverse problem , residual , curve fitting , tikhonov regularization , mathematical optimization , mathematical analysis , algorithm , geometry , computer science , statistics , artificial intelligence
We attempt to select an optimal value of regularization parameter in the optimization problems for intensity‐modulated radiotherapy which are solved using a variational regularization technique. We apply to inverse treatment planning the L‐curve method which was developed to determine the regularization parameter in the discrete ill‐posed problems. The L‐curve method is based on finding the regularization parameter which minimizes the residual norm which is a measure of accuracy of fit and the solution norm which is a measure of smoothness of solution. The main idea of the L‐curve method is to plot the smoothing norm as a function of the residual norm for all values of the regularization parameter. This characteristic curve has an L‐shaped dependence and the optimal value of regularization parameter can be found at the “corner” of the L‐curve. We plot the L‐curves for the optimization problems which simulate prostate radiotherapy cancer treatment with intensity‐modulated beams. Different numerical methods are applied to calculate the point of maximum curvature of the L‐curves which is a criterion to locate the corner. We show that the point of maximum curvature can be located in a most robust way using a formula derived from the singular value decomposition analysis.